Question

find the zeros of the polynomial and state the multiplicity: f(x)=x^3-x^2-4x+4

Answer 1

How do you find the possible rational zeros?

Answer 2

that's what I don't understand :/

Answer 3

Can you answer these questions?
Find the factors of the constant term.
Find the factors of the coefficient of the term with the highest exponent.

Answer 4

Find the factors of the constant term means find the factors of 4 since 4 is the constant term.
So what are the factors of 4?

Answer 5

1, 2, and 4?

Answer 6

well plus or minus but okay.
Now what are the factors of the coefficient of the term with the highest exponent.
This means I'm looking at 1 since 1 is in front of x^3.
What are the factors of 1?

Answer 7

1

Answer 8

well plus or minus but yeah

Answer 9

oh okay so +/- 1

Answer 10

We define the possible rational zeros as the set containing the factors of the constant term/ the factors of the coefficient of the term with highest exponent.

So this means we have \[\text{ possible rational zeros } = \{ {\frac{\pm 1}{\pm 1} , \frac{\pm 2}{\pm 1}, \frac{\pm 4}{\pm 1} }\}\]

Answer 11

or just simply \[\{ \pm 1 , \pm 2 , \pm 4\} \]

Answer 12

Now possible doesn't mean actual.

Answer 13

This just narrows are search.

Answer 14

So we have f(x)=x^3-x^2-4x+4 .
We try some of our possible rational zeros to see which gives us 0.

Answer 15

What value in our possible rational set do you want to try and see if it works?

Answer 16

would we do 1 since it is first?

Answer 17

You can but I was asking you where you want to start.

Answer 18

So if we plug in 1 do we get 0?

Answer 19

am I plugging 1 in for x?

Answer 20

We are trying to find for what values of x is f zero so yes.

Answer 21

okay by plugging 1 in for x it does equal 0

Answer 22

\[f(1)=1^3-1^2-4(1)+4 \]
is this zero?
You are right the answer is yes.
x=1 is a zero since f will be 0 when x=1
If x=1 is a zero then x-1 is a factor of f.
Now divide f by x-1.
You may use synthetic or long polynomial division.

Answer 23

what would my equation be for that? do I just plug in x-1?

Answer 24

for f?

Answer 25

like you are dividing f by x-1 to find the other factors of f.

Answer 26

remember f is x^3-x^2-4x+4

Answer 27

f/(x-1)=?

Answer 28

\[\frac{x^3-x^2-4x+4}{x-1} =?\]

Answer 29

\[0+ \frac{ 4x }{ x-1 }\]

Answer 30

No that is way off. I'm sorry.

Answer 31

Do you know how to do long division with polynomials?

Answer 32

I just don't understand how to do this I suppose

Answer 33

Have you discussed long division?
I could show you an easier way but doesn't always work out so well like polynomial division does.

Answer 34

I tried to do the problem in synthetic division but it just said it's not possible

Answer 35

it is possible